ON THE POINCARÉ CONSTANT OF LOG-CONCAVE MEASURES

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincaré constant. First we revisit E. Milman's result [40] on the link between weak (Poincaré or concentration) inequalities and Cheeger's inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincaré (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the logconcave case, to reduce to the case of the Poincaré inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...)

Economic Consequences of Current Precarious Employment, Alignment Paths and Development Trends

The economic crisis associated with the epidemiological situation in the other countries and in Russia in particular has a powerful impact on the level of employment now, and further, according to analysts, the situation will worsen. The unemployment rate is rising, along with the level of employment falling. The role of small business is not only to maintain balance in the labor market, create new jobs and absorb excess labor, but also to develop the economy and ensure the revenue of the state budget. However, to optimize their costs, employers go into the shadows. This problem has always existed; tax control measures have made it possible to solve it for the formal sector of the economy, though it still remains among small businesses. There is a need to develop, improve the efficiency of employment as well as research and optimization of precarious employment. The paper analyzes the category of “precarious employment” conducting a study of both employment and precarious employment in various businesses, identifies promising areas against precarization of employment. Proposals have been made to determine the category of precarious employment in small businesses, to increase the level of employment in certain sectors over the long term

Modifications of a method for low energy gamma-ray incident angle reconstruction in the GAMMA-400 gamma-ray telescope

The GAMMA-400 gamma-ray telescope is designed to measure the gamma-ray fluxes in the energy range from 3c20 MeV to 3c1 TeV, performing a sensitive search for high-energy gamma-ray emission when annihilating or decaying dark matter particles. Such measurements will be also associated with the following scientific goals: searching for new and studying known Galactic and extragalactic discrete high-energy gamma-ray sources (supernova remnants, pulsars, accreting objects, microquasars, active galactic nuclei, blazars, quasars). It will be possible to study their structure with high angular resolution and measuring their energy spectra and luminosity with high-energy resolution; identify discrete gamma-ray sources with known sources in other energy ranges. The major advantage of the GAMMA-400 instrument is excellent angular and energy resolutions for gamma rays above 10 GeV. The gamma-ray telescope angular and energy resolutions for the main aperture at 100-GeV gamma rays are 3c0.01% and 3c1%, respectively. The motivation of presented results is to improve physical characteristics of the GAMMA-400 gamma-ray telescope in the energy range of 3c20-100 MeV, most unexplored range today. Such observations are crucial today for a number of high-priority problems faced by modern astrophysics and fundamental physics, including the origin of chemical elements and cosmic rays, the nature of dark matter, and the applicability range of the fundamental laws of physics. To improve the reconstruction accuracy of incident angle for low-energy gamma rays the special analysis of topology of pair-conversion events in thin layers of converter performed. Choosing the pair-conversion events with more precise vertical localization allows us to obtain significantly better angular resolution in comparison with previous and current space and ground-based experiments. For 50-MeV gamma rays the GAMMA-400 gamma-ray telescope angular resolution is better than 50